Problem
GEO-B2-M10-P009 A Transversal with an Exterior Point
#9
★★★☆☆ Level 3 of 5
In triangle \(ABC\), point \(D\) lies on \(AB\), point \(E\) lies on \(BC\), and point \(F\) lies on the extension of \(CA\) beyond \(A\). Suppose \(AD:DB=1:2\), \(BE:EC=2:3\), \(AF:FC=1:3\). Prove that \(D,E,F\) are collinear.
For collinearity of points on sides and one extension, apply Menelaus.
Since \(F\) lies on the extension beyond \(A\), \(CF:FA=3:1\). Check Menelaus' product: \(\frac{AD}{DB}\cdot\frac{BE}{EC}\cdot\frac{CF}{FA}=\frac{1}{2}\cdot\frac{2}{3}\cdot 3=1\). Hence \(D,E,F\) are collinear.
The student must notice the exterior point and not apply Ceva.